Even though the operator–sum representation gives an “internal” view of the dynamics of an open…

Even though the operator–sum representation gives an
“internal” view of the dynamics of an open system, we can still learn something
about the environment from the Ak operators. For a given evolution
map E and an input state _, define Wkl by

Show that these numbers are the matrix elements of the final
density operator of the environment E, if the initial state of E is pure (as in
Eq. 9.1).

To make this more definite, suppose the Kraus operators for
a qubit system are a1 and bZ, where a and b are real parameters such that a2
+ b2 = 1. First, confirm that this pair of operators is
»

Even though the operator–sum representation gives an
“internal” view of the dynamics of an open system, we can still learn something
about the environment from the Ak operators. For a given evolution
map E and an input state _, define Wkl by

Show that these numbers are the matrix elements of the final
density operator of the environment E, if the initial state of E is pure (as in
Eq. 9.1).

To make this more definite, suppose the Kraus operators for
a qubit system are a1 and bZ, where a and b are real parameters such that a2
+ b2 = 1. First, confirm that this pair of operators is
normalized according to Eq. 9.12. Then find the eigenvalues of the final
environment state when the input qubit states are  

»

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